Padé approximation of model with time delays

`[num,den] = pade(T,N) `

pade(T,N)

sysx = pade(sys,N)

sysx = pade(sys,NU,NY,NINT)

`pade`

approximates time delays by rational
models. Such approximations are useful to model time delay effects
such as transport and computation delays within the context of continuous-time
systems. The Laplace transform of a time delay of *T* seconds
is exp(–*sT*). This exponential transfer function
is approximated by a rational transfer function using Padé approximation
formulas [1].

`[num,den] = pade(T,N) `

returns the Padé approximation
of order `N`

of the continuous-time
I/O delay exp(–*sT*) in transfer function
form. The row vectors `num`

and `den`

contain
the numerator and denominator coefficients in descending powers of *s*.
Both are `N`

th-order polynomials.

When invoked without output arguments, `pade(T,N)`

plots
the step and phase responses of the `N`

th-order Padé
approximation and compares them with the exact responses of the model
with I/O delay `T`

. Note that the Padé approximation
has unit gain at all frequencies.

`sysx = pade(sys,N) `

produces
a delay-free approximation `sysx`

of the continuous
delay system `sys`

. All delays are replaced by their `N`

th-order
Padé approximation. See Time Delays in Linear Systems for more information
about models with time delays.

`sysx = pade(sys,NU,NY,NINT)`

specifies independent approximation
orders for each input, output, and I/O or internal delay. Here `NU`

, `NY`

,
and `NINT`

are integer arrays such that

`NU`

is the vector of approximation orders for the input channel`NY`

is the vector of approximation orders for the output channel`NINT`

is the approximation order for I/O delays (TF or ZPK models) or internal delays (state-space models)

You can use scalar values for `NU`

, `NY`

,
or `NINT`

to specify a uniform approximation order.
You can also set some entries of `NU`

, `NY`

,
or `NINT`

to `Inf`

to prevent approximation
of the corresponding delays.

High-order Padé approximations produce transfer functions
with clustered poles. Because such pole configurations tend to be
very sensitive to perturbations, Padé approximations with order `N>10`

should
be avoided.

[1] Golub, G. H. and C. F. Van Loan, *Matrix
Computations*, Johns Hopkins University Press, Baltimore,
1989, pp. 557-558.

`absorbDelay`

| `c2d`

| `thiran`

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